Chapter 5, Harder Questions, Answers
Q1
Does the termab initioimply that such calculations are “exact”? In what sense
mightab initiocalculations be said to be semiempirical – or at least not a priori?
The term does not imply that such calculations are exact. This is clear from the
fact that most ab initio calculations use an approximate Hamiltonian, and all use a
finite basis set.
The Hamiltonian: In noncorrelated calculations the main error in the Hamilto-
nian is that it does not take electron correlation into account properly, treating it in
an average charge-cloud way. But even in correlated calculations the Hamiltonian
can contain approximations: it is usually nonrelativistic, which introduces signifi-
cant errors for heavy atoms, and it routinely ignores spin-orbit coupling (spin-
orbit interaction), which can be important [1a]. There are still other effects,
usually small and rarely taken into account: spin–spin interaction between elec-
trons [1b], neglect of the finite sizeof the nuclei [1c], and the use of the
Born–Oppenheimer approximation [2]. The point is not that these effects are
necessarily important, but that their neglect renders the calculation, strictly
speaking, inexact.
The basis set: using a finite basis set necessarily leads to an inexact wavefunc-
tion, in much the same way that representing a function by a finite Fourier series of
sine and cosine functions necessarily gives an approximation (albeit perhaps an
excellent one) to the function.
Dewar,thepioneer of the semiempirical approach and a longtime critic of
shortcomings (sometimes perhaps viewed with too jaundiced an eye) of ab initio
calculations, gives an account of the origin of the term [3]. Although I cannot vouch
for the definitive accuracy of this anecdote, it is amusing:
The term “ab initio” was originally applied to the Roothaan–Hall (RH) approach though an
amusing accident. Parr was collaborating in some work of this kind with a group in
England. In reporting one of his calculations he described it as “ab initio”, implying that
the whole of that particular project had been carried out from the beginning in his
laboratory. The term, unfortunately, became generally adopted for all calculations of this
kind...
None of the above caveats should be taken to imply that excellent results cannot
be obtained from ab initio calculations. However, except perhaps for calculations at
so high a level that they are essentially exact solutions of the Schr€odinger equation,
one must use experiments on related systems as a reality check, as was emphasized
by Dewar [3]. It is in this sense that ab initio calculations are semiempirical (in fact,
in the literature they are never really described as such): not at all in the sense that
they are parameterized against experiment, but in the sense that for justified
confidence in their results one should (almost always) check representative calcula-
tions against reality.
Answers 621